# Setting up a local-coordinate system in space-time using only a single clock and light beams

I have a question to ask about the operationalist view of space-time. I am a mathematician who happens to be interested in physics, so if anyone thinks that my question is a silly or vague one, please feel free to close it.

Many physics textbooks that discuss Special Relativity mention that one way of setting up a coordinate system in flat Minkowski space is to imagine that there is an infinite grid of meter rules that pervades the universe, with synchronized clocks located at each node of the grid. In order to obtain more precise measurements of space-time events, one simply divides the grid into more nodes. I feel that this presents problems, because what does it mean to say that ‘all the clocks are synchronized’? Any act of moving a clock from a starting point to its designated node is sure to involve some acceleration, and this is going to upset any synchronization attempt. Also, such a description is not satisfactory because it is impossible to deal with an infinite number of meter rules and clocks.

Things are made worse in curved space-time because of the further effect of gravitational time-dilation. Hence, I believe that the only logically consistent way to perform measurements of space-time events is to use a single atomic clock carried by a single observer and to use the world-lines of photons emitted by a light source carried by the observer to ‘carve out’ a local-coordinate system. The exact description of how this can be done is beyond me, so I would like to gather some useful information from anyone in the community. Thank you very much!

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Hi, and welcome to Physics Stack Exchange! Are you asking specifically whether it is possible to set up a global coordinate system using a single clock, or would you be interested in a description of how clocks at separate locations can be synchronized? –  David Z Feb 1 '13 at 5:43
Answers to both questions would be greatly appreciated, but I am more interested in the first question. Also, as I am interested in curved space-time, I suppose that we can only talk about local-coordinate systems. I know that many philosophical questions can be answered using light beams, such as what it means for a curve to be ‘straight’, but I would like to see how one can use light beams and a single clock to set up, in a logically consistent way, a local coordinate system in order to perform measurements, all in a manner that conforms to the operationalist view of space-time. –  Haskell Curry Feb 1 '13 at 5:55

The radar method is a general approach that works for non-inertial observers and curved spacetime.

Two co-ordinates of an event are given by your clock time at which the event intersects your future and past light cone, called retarded time and advanced time, ($\tau^+,\tau^-$, resp.). Or use a diagonal combination thereof: $\tau^\star = \frac{\tau^+ + \tau^-}{2}$, called radar time, and $\rho = c\frac{\tau^+ - \tau^-}{2}$, called radar distance.

This diagonal combination has the property that, in the case of an unaccelerated observer in flat spacetime, $\tau^\star$ and $\rho$ are equal to the usual measures (the "infinite grid of rulers and clocks" business).

Two other co-ordinates can be given by the incoming angles ($\Omega^+$) of the null geodesic from the event to you. This is the reception or retarded co-ordinate system. The dual system, the trasmission or advanced co-ordinate system, would use the outgoing angles ($\Omega^-$) of the null geodesic from you to the event.

For a non-rotating unaccelerated observer in flat spacetime, the two pairs of angles are equal to each other and to the usual polar co-ordinate angles.

In flat spacetime this will assign a unique co-ordinate to every reachable event, that is, every event in the observer's causal diamond. In curved spacetime it will assign at least one co-ordinate to every reachable event, however there may be duplicates. One can restrict to the boundary of the causal past and future, as described in answer I linked to above. Then, under certain causality assumptions, every reachable event gets a unique $\tau^\star$ and $\rho$. The surfaces of constant $\tau^\star$ and $\rho$ are then 2-D globally spacelike surfaces, but not always topologically $\mathcal{S}^2$, rather, they will be some subquotient of $\mathcal{S}^2$. That is, for a given $\tau^\star$ and $\rho$ some angle pairs $\Omega^+$ will not be valid (corresponding to parts of the light cone that have "fallen behind"), and some events on the boundary of validity will have more than one angle pair.

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Thank you, Retarded Potential! The link that you’ve provided contains another link to just the paper that I’m looking for. –  Haskell Curry Feb 3 '13 at 1:30

The idea of the of the coordinate system in Minkowsky space is a generalization of, for example, a Cartesian coordinate system in which at any point you have a set of numbers that give your position within that coordinate system. The clocks are added to be able to visualize the coordinate system in 3 dimensions but you can also think as time being a perpendicular axis to the other 3.

For example imagine in 1+1 dimensions, that's 1 space dimension and 1 time dimension. You extend an infinite one dimensional rod and place clocks on it, with a spacing that can be as small as you want. Then if you are on the rod at some space point and time, you get a number from the rod which represents your space coordinate, and a number from the clock which is your time coordinate. The clocks are synchronized in the sense that if you send a beam of light from a point $(x_0,t_0)$, when the beam reaches a space position of an observer at $x_0+\Delta x$, the clock beside the observer will measure a time $t_0+\Delta x/c$.

Another way of looking at this construction is embedding it in a 2D Euclidean space, in which some straight infinite line represents the space coordinate and a perpendicular straight line represents the time coordinate. In this representation, it is clearer what we mean by having synchronized clocks. It is just that the path in which light travels in this 2D representation has a slope of 45º (assuming $c=1$). Thus, in the clock-rod picture, dividing the grid into more nodes means, in this picture, that you add more cuts on your perpendicular axes. This shouldn't cause more trouble than constructing, for example, the real line (which for some pure mathematicians has some logical flaws).

On the second note, about curved space time. The word curved means that in the clock-rod picture you cannot extend your rods forever because of the intrinsic curvature of space-time. What you can do then, is extend them only to a certain extend, which you consider as a "local" coordinate system. Thus this coordinate system is only valid in a neighborhood of some space-time point you chose. This construction is realized naturally on manifolds, as in those mathematical structures you have a notion of a local coordinate system associated to every element of your topological space.

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(Note: Please see addendum at bottom for paper references relevant to the question.)

The question of how to synchronized clocks was first addressed by Einstein himself in his famous special relativity paper, and it occupies much of the discussion in that paper prior to him getting into electromagnetics.

What Einstein suggested was this: Both observers have identically constructed clocks, and from that they assume that the clocks keep equal time. But how do they make sure they have the same time, not just the same tick rate?

Einstein suggested what would now be called a handshake: A sends a time stamp to B via a light beam, and B immediately sends his own time stamp back to A. When A receives that "handshake" back from B, she knows both how long it took to get the information and what time B had (by symmetry) halfway through that time period.

That's enough information for A to update her clock to that of B, plus half of the delay in the response to take care of how long it took B's time stamp to arrive.

There are other more subtle issues such as whether B might be moving relative to A, but those too can be taken care of using only light beams by ensuring there is no frequency shift (Doppler effect) when viewing the returning beam.

Not only is this procedure pretty straightforward, it's useful. For example, you would not be reading this message if the electric company you use didn't use the same kind of synchronization procedure to ensure that distant parts of an electrical network are all very precisely in sync with each other. If they did not do that, the generators would get out of phase and start destroying each other. Meaningful time synchronization thus is not some abstract concept, but something real and very much needed for anything networked. The important point for getting this type of synchronization in time is that the various parts must not be moving relative to each other. That's where special relativity kicks in, not in synchronization itself.

There is another technique that you touched on, which is this: If you move a clock very slowly from one location to another, the relativistic effects can be made very, very small -- so small in fact that you really can always make them small enough to ignore them. It's not nearly as practical as Einstein synchronization, but it comes in handy at times.

Now, as for dividing up the rulers, yes, you cannot get infinitely small. But you can actually do very well at it simply by creating networks of very small clocks. Many computer networks are actually pretty decent examples of that. However, my favorite image is that of an entire region of space filled with tiny, tiny clocks, say a millimeter across each, with each clock constantly synchronizing with its nearest neighbors.

As Einstein himself pointed out, synchronization is a transitive operation, so having all those clocks talking to each other in that fashion eventually leads to an entire region of space filled with meaningfully synchronized particle-like clocks. Add a bit of data recording, and you can also use that network of particle-like clocks to keep that region of space both highly synchronized and capable of collecting data about both itself and other objects.

Even more interesting, if another object goes sailing through such a region at high speed, it can in principle extract a very exact time from each of the particle-like clocks with which it collides. The time is exact because the contact is "proximate" or touching, which eliminates the usual spacetime ambiguities when exchanging data between frames. The fast-moving object thus can "read" exactly what time the network of particle-like clocks thinks it is at each point of its journey, and vice-versa.

Surprisingly, and not very obviously, even this concept is in Einstein's paper after a fashion, for this reason: He invokes the idea of a rod moving through another frame and "reading" the time of that frame at each end of the rod. As it turns out, the only way you can do that is by creating a synchronized network of very tiny clocks that can "touch" each end of the rod as it passes. Any scheme that uses light (versus proximity) to pass the same information immediately becomes ambiguous, since different frame would interpret the message passing as different "mixes" of space and time. Proximate contact removes that, and enable Einstein's original moving-rod though experiment to be realizable in real space with real equipment.

I have never seen a name for the idea of asymptotically shrinking, particle-like, data-collecting, fully networked and fully synchronized nano-clocks occupying a volume of space, but it's a straightforward and definitely doable extension of things that modern communications systems do all the time. I like to call this idea of nano-sized clocks a synchronized network of observing particles, for which the acronym is, well... snoop.

So, if you want to do Einstein's moving rod experiment in real life, you will pretty much have to create some form of snoop first. There are actually much easier ways to do it than building real nano-clocks -- cloud chambers with carefully timed imaging certainly come to mind -- but the concept of a snoop helps analyze how you would go about doing many somewhat obscure-sounding special relativity experiments without getting lost in ambiguous data.

Please notice that I've intentionally gone in a somewhat different direction from what I think your suggestion is, which is to use a single clock with photon world lines extending out from it. The single central clock works great for defining a single cell of space and time, but I think you would find that you would end up with just one large-granularity pendulum clock by the time you work out all the details of the idea. You can do a lot with one well-defined cell of that type, but the fact that you can't even do Einstein's very first special relativity thought experiment (the moving rod) requires multiple synchronized clock cells is a pretty good argument that synchronization and multiple cells (multiple identical clocks) will also be needed to collect enough data to test even special relativity in detail, let along general.

If on the other hand you accept the idea of a "snoop limit" at which you can approximate synchronized clocks at point in a region of space to whatever level of detail is sufficient for your particular experiment, I think you'll end up with a much more satisfying (and experimentally unambiguous) result. In terms of those light pendulums I mentioned:

... you network a large number of them together by having them touch on their side corners. If the central clocks can also collect data, that's actually the more precise way to define a snoop.

Addendum 2013-02-02: Several relevant online references

As a result of the excellent information added via @RetardedPotential's answer, I now know the "standard" name for rho cells within the curved-space community: causal diamonds. The beautiful causal diamond diagram in the link I just gave is from this 2009 blog entry by theoretical physicists Sabine Hossenfelder and Stefan Scherer. From the variety I'm seeing in searching for examples, there does not seem to be any single highly standardized way of labeling the backward and forward light geodesics that form the sides of causal diamonds. I labeled them ${\pi_\phi}^-$ and ${\pi_\phi}^+$ for the backwards ($-$) and forwards ($+$) photon coordinates ($\pi$) of frame $\phi$, but I may relabel them if something more standard exists.

More importantly, the idea of using cells small enough to ensure that space is locally flat -- the same smoothness assumption that underlies calculus -- has also been explored! I call such networks snoops, but the 2007 G.W. Gibbons and S.N. Solodukhin paper The geometry of small causal diamonds (Physics Letters B, Vol. 649, Issue 4, 7 June 2007, p.317–324) clearly recognizes this same idea and explores it from the perspective of analyzing curved space.

And wow! Here's what looks to be a very relevant to fine-grained causal diamond analysis from just a couple of months ago:

The Discrete Geometry of a Small Causal Diamond, by Mriganko Roy, Debdeep Sinha, and Sumati Surya. Submitted to arXiv on on 4 Dec 2012

So, Haskell Curry, you may want to check out the above references as starting points if you are interested in fine-grained mapping of curved spaces.

(My own SR interests in snoops remain a bit tangential to curved spaces... um, did I just make a pun?)

2013-02-03

Back in 2007, @lubosmotl summarized a talk by Raphael Bousso in which Bousso mentions causal diamonds in the context of his holographic universe idea. The phrase is clearly older, e.g. see this 1999 mention of causal diamonds by George Svetlichny. Most likely it was a common catch-phrase in the 1990s for the straightforward relativity concept of intersecting forward and backward light cones.

My rho cells necessarily have the same geometry as causal diamonds, but that's about as far as the resemblance goes. By definition a rho cell has an unaccelerated rest-mass clock as its spine, and uses that clock to define $\tau_\rho$, $t_\rho$, and $l_\rho$, the last being isotropic only within clock frame. The clock could be as simple as a muon, and the reflectors could be replaced by photon exchanges between clocks. However, without a rest-mass spine, I don't readily see how a causal diamond can have a cause to be causal about, so to speak.

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Terry, thank you for a really informative explanation. I still have some questions left. Your description of clock synchronization fits neatly into the operational view of space-time, as it can be carried out physically (I like the idea of a time-stamp). However, this makes sense only in flat space-time. In curved space-time, however, how does one carry out measurements in a valid manner? Any region of curved space-time, no matter how small, is ultimately still curved. Can a huge collection of clocks in a volume of curved space-time yield direct information about the curvature itself? –  Haskell Curry Feb 1 '13 at 9:13
For example, can a concentrated collection of nano-clocks allow us to compute the metric tensor of curved space-time? Also, it seems that clocks are doing most of the work, and the role of light beams is simply to relay information between clocks. This sounds absolutely fine to me, as long as it gives us a meaningful way to make measurements, even in regions of space-time with high curvature, such as near a black hole. –  Haskell Curry Feb 1 '13 at 9:17
Haskell Curry, sorry for the delays, was on travel. The volume-of-clock methods use the same smoothness assumptions as calculus: It is assumed that for any given situation you can make the clock lattice sufficiently small for the local curvature to become vanishingly small. Curvature then becomes an explicit metric of how you perform the synchronization process, with non-trivially curved space requiring e.g. that clocks in gravity wells really will be slower than others. So the point was just that: snoops allow detailed analysis of geodesics, at least in principle. –  Terry Bollinger Feb 3 '13 at 2:00
Also, I should note: Einstein's original synchronization method of course assumes flat space, since he wrote it a decade before his general relativity paper! But it's more general than that, if you make the clocks explicit and require that the network as a whole settle into a single non-oscillatory solution. A rather interesting problem, that. Also: At a quick read, Retarded Potential (I still chuckle every time I read that name) has very nicely captured the curvature issues for a large cells -- causal diamonds? is that what they are called? Cool! -- while I was going for small granularity. –  Terry Bollinger Feb 3 '13 at 2:10
@RetardedPotential, thanks! I did not have that keyword phrase. The diamonds are of course really intersecting conics in 4D. Apparently what I called snoops have been analyzed... ah, here's a good one. The free arXiv version is: G.W. Gibbons and S.N. Solodukhin, "The geometry of small causal diamonds." Physics Letters B, Vol. 649, Issue 4, 7 June 2007, p.317–324 Six years ago even! Anyway, if you are still interested in fine-grained space analysis, that looks like a great connection point. I'm intrigued and will investigate more. –  Terry Bollinger Feb 3 '13 at 2:36